Problem: Points $A,$ $B,$ $C,$ and $D$ are equally spaced along a line such that $AB = BC = CD.$  A point $P$ is located so that $\cos \angle APC = \frac{4}{5}$ and $\cos \angle BPD = \frac{3}{5}.$  Determine $\sin (2 \angle BPC).$
Solution: Let $a = AP,$ $b = BP,$ $c = CP,$ and $d = DP.$  Let $\alpha = \angle APC,$ $\beta = \angle BPD,$ $\gamma = \angle BPC,$ and $\delta = \angle APD.$  Then $\cos \alpha = \frac{4}{5}$ and $\cos \beta = \frac{3}{5}.$  Since
\[\cos^2 \alpha + \cos^2 \beta = 1,\]and $\alpha$ and $\beta$ are acute, these angles must satisfy $\alpha + \beta = 90^\circ.$  Also, $\sin \angle APC = \frac{3}{5}$ and $\sin \angle BPD = \frac{4}{5}.$

[asy]
unitsize (2 cm);

pair A, B, C, D, P, Q, R;

A = (0,0);
B = (1,0);
C = (2,0);
D = (3,0);
Q = (1,3);
R = (2,2);
P = intersectionpoints(circumcircle(A,Q,C),circumcircle(B,R,D))[0];

draw(A--D);
//draw(circumcircle(A,Q,C));
//draw(circumcircle(B,R,D));
draw(A--P--D);
draw(P--B);
draw(P--C);
draw(arc(P,0.3,degrees(A - P),degrees(C - P)),red);
draw(arc(P,0.5,degrees(B - P),degrees(D - P)),red);
draw(arc(P,0.6,degrees(B - P),degrees(C - P)),red);
draw(arc(P,0.9,degrees(A - P),degrees(D - P)),red);

label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, SE);
label("$P$", P, N);
label("$a$", interp(A,P,0.2), NW, red);
label("$b$", interp(B,P,0.2), NW, red);
label("$c$", interp(C,P,0.2), W, red);
label("$d$", interp(D,P,0.2), E, red);
label("$\alpha$", P + (-0.25,-0.35), UnFill);
label("$\beta$", P + (-0.05,-0.65), UnFill);
label("$\gamma$", P + (-0.35,-0.7), UnFill);
label("$\delta$", P + (-0.45,-0.95), UnFill);
[/asy]

Note that triangles $ABP,$ $BCP,$ and $CDP$ have the same base and height, so their areas are equal.  Let $K = [ABP] = [BCP] = [CDP].$

We have that
\[[APC] = \frac{1}{2} ac \sin \angle APC = \frac{3}{10} ac,\]so $K = \frac{1}{2} [APC] = \frac{3}{20} ac.$

Also,
\[[BPD] = \frac{1}{2} bd \sin \angle BPD = \frac{2}{5} bd,\]so $K = \frac{1}{2} [BPD] = \frac{1}{5} bd.$  Hence,
\[K^2 = \frac{3}{100} abcd.\]Also,
\[[APD] = \frac{1}{2} ad \sin \delta,\]so $K = \frac{1}{3} [APD] = \frac{1}{6} ad \sin \delta.$  Since $K = [BPC] = \frac{1}{2} bc \sin \gamma,$
\[K^2 = \frac{1}{12} abcd \sin \gamma \sin \delta.\]It follows that
\[\sin \gamma \sin \delta = \frac{9}{25}.\]Note that $\gamma + \delta = \alpha + \beta = 90^\circ,$ so $\delta = 90^\circ - \gamma.$  Then $\sin \delta = \sin (90^\circ - \gamma) = \cos \gamma,$ and
\[\sin \gamma \cos \gamma = \frac{9}{25}.\]Therefore, $\sin 2 \gamma = 2 \sin \gamma \cos \gamma = \boxed{\frac{18}{25}}.$